Digital wireless communications are being widely used throughout the world particularly with the latest development of the Orthogonal Frequency Division Multiplex (OFDM systems) and the latest evolution, namely the so-called Long Term Evolution (LTE), DVB-H, WiFi 802.11 and WiMax 802.16 systems.
OFDM is a frequency-division multiplexing (FDM) scheme utilized as a digital multi-carrier modulation method. As it is well—known to one skilled in the art, OFDM systems demonstrate significant advantages in comparison to single-carrier schemes, particularly in their ability to cope with severe channel conditions (i.e. channel attenuation, narrowband interference, frequency-selective fading).
However, OFDM systems performances are significantly impaired by the rising of inter-carrier interference (ICI) effect in the presence of time-varying propagation channels. Those circumstances occur in mobile cellular OFDM communication systems as those envisaged in the above mentioned standards.
More specifically, the movement of a User Equipment (UE) in high velocity results to high Doppler spread and accordingly to fast time-varying propagation channels. In turn, the resulting fast time-varying propagation channels yield to significant ICI. In practice, the increased ICI prevents classical OFDM receiver schemes, and particularly the so-called matched filter model—from reliably detecting the desired signal. Hence more advanced receiver equalization techniques are required to mitigate the effect of the ICI.
A known solution for reducing the ICI is based on the performance of a full matrix (N×N) inversion where N represents the OFDM Fast Fourier Transform (FFT) order by Gaussian Elimination. However, the high complexity (O(N3)) of achieving such an inversion cannot be supported by a practical mobile receiver.
To solve the problem of high complexity iterative techniques have been widely used in the art in order to avoid a full matrix inversion. These techniques are based in solving linear systems of equations [1]. Considering a generic linear system of equations of the formAx=b where the vector x is the sequence to be estimated, b is the observation vector, and the matrix A (N×N) is a full-rank input-output transfer matrix, then for any iterative estimation method, the convergence of the sequence estimates (κ)→ is governed by the spectral properties of the matrix A. A commonly used metric for those spectral properties is the condition-number κ(A), defined as the ratio between the largest and smallest eigen-values of A, κ(A)=|λmax(A)/λmin(A)| (see reference [1]). Specifically, the closer condition number (CN) is to 1, the faster a given iterative algorithm will converge.
Clearly, the convergence of an algorithm has a significant impact on the performance of the receiver since, firstly, a fast converging algorithm saves the battery life of the receiver (in the case of a mobile receiver) and, secondly, a fast convergence algorithm allows to comply more easily to the requirements given by the standards.
Furthermore, iterative techniques can greatly take advantage from appropriate preconditioning to reduce the CN and to allow faster convergence. Specifically, the iterative methods are applied in an equivalent preconditioned linear system derived from the above mentioned linear system of equations intoPAx=Pb with P being the preconditioning matrix and such that κ(A)≧κ(PA)≧1 with PA=1 if P−1=A.
In prior art there are many preconditioning techniques [1]. Among those, a simple and straightforward technique is the Jacobi preconditioning. Specifically, in Jacobi preconditioning, P is chosen to be diagonal and such that diag{P−1}=diag{A} if [A]ii≠0 for i=1, . . . , N.
The Jacobi preconditioning consists in approximately solving the problem of inverting matrix A and transform the original problem into a better conditioned one.
Although Jacobi preconditioning significantly reduces the complexity and time required for achieving full matrix A inversion, there is still a need for more advanced preconditioning techniques for achieving the same.